Gauss' trigonometric identities for heptadecagon

Today, we write down Gauss' magical trigonometric identities for regular heptadecagon.

James' geometry art

Construction algorithm

In our previous post, we have learnt about Similar Triangles and the Right Triangle Altitude Theorem. Today, continuing our journey in the garden of geometry, we want to find answer to the following question

Given a line segment of length $r$, by compass and straightedge, what kind of shapes can we construct?